Deconvolution derivation

There is a large body of literature on FTIR spectroscopy including, for example, Bracewell (1965), Horlick (1968), Bell (1972), Griffiths (1975, 1983), Ferraro and Basile (1978), Koenig (1981), Griffiths and de Haseth (1986), Perkins (1986, 1987), Mackenzie (1988). Cameron and Moffatt (1984), and Gillette et al. (1985) have explained the basic mathematical concepts of deconvolution, derivation and smoothing in FTIR spectroscopy. 

In the spectra of macromolecules, the overlapping bands cannot be resolved by instrumental manipulation alone. Instead, mathematical techniques such as deconvolution, derivative spectroscopy, and band fitting are used for improving the spectral quality. 

Cameron DG, Moffatt DJ (1984) Deconvolution, derivation, and smoothing of spectra using Fourier transforms Test Eval 76 83... 

Methods for evaluation of analytical signals are as follows transformation, smoothing, correlation, convolution, deconvolution, derivation, and integration. 

Several mathematical algorithms have been developed to speed up the Fourier transformation of discrete data. The most successful of these has been the fast Fourier transform (FFT) derived by Cooley and Stukey. There are also inverse FFTs (IFFTs) to convert from frequency data to the time domain. IFFTs are mostly used in mathematical manipulations of data, such as data reduction, deconvolutions, derivatives, etc. 

Figure 2.36 A shows a typical low-loss spectrum taken from boron nitride (BN). The structure of BN is similar to that of graphite, i. e. sp -hybridized carbon. For this reason the low-loss features are quite similar and comprise a distinct plasmon peak at approximately 27 eV attributed to collective excitations of both n and a electrons, whereas the small peak at 7 eV comes from n electrons only. Besides the original spectrum the zero-loss peak and the low-loss part derived by deconvolution are also drawn. By calculating the ratio of the signal intensities hot and Iq a relative specimen thickness t/2 pi of approximately unity was found. Owing to this specimen thickness there is slight indication of a second plasmon.

However, the use of a HPLC separation step enabled a remarkable acceleration of the deconvolution process. Instead of preparing all of the sublibraries, the c(Arg-Lys-O-Pro-O-P-Ala) library was fractionated on a semipreparative HPLC column and three fractions as shown in Fig. 3-2 were collected and subjected to amino acid analysis. According to the analysis, the least hydrophobic fraction, which eluted first, did not contain peptides that included valine, methionine, isoleucine, leucine, tyrosine, and phenylalanine residues and also did not exhibit any separation ability for the tested racemic amino acid derivatives (Table 3-1). 

The improvements in resolution achieved in each deconvolution step are shown in Figure 3-3. While the initial library could only afford a modest separation of DNB-glutamic acid, the library with proline in position 4 also separated DNP derivatives of alanine and aspartic acid, and further improvement in both resolution and the number of separable racemates was observed for peptides with hydrophobic amino acid residues in position 3. However, the most dramatic improvement and best selectivity were found for c(Arg-Lys-Tyr-Pro-Tyr-(3-Ala) (Scheme 3-2a) with the tyrosine residue at position 5 with a resolution factor as high as 28 observed for the separation of DNP-glutamic acid enantiomers. 

Peptoid hbraries have also yielded compounds active in the disruption of RNA-protein interactions. Compounds not derived from library syntheses are discussed in Section 1.4.1. A peptoid 9-mer with a number of cationic groups was discovered (Fig. 1.4) after several rounds of mixture deconvolution, that was able to block the interaction of HIV-1 Tat protein with TAR RNA at nanomolar concentra-... 

It is still possible to enhance the resolution also when the point-spread function is unknown. For instance, the resolution is improved by subtracting the second-derivative g x) from the measured signal g x). Thus the signal is restored by ag x) - (7 - a)g Xx) with 0 < a < 1. This llgorithm is called pseudo-deconvolution. Because the second-derivative of any bell-shaped peak is negative between the two inflection points (second-derivative is zero) and positive elsewhere, the subtraction makes the top higher and narrows the wings, which results in a better resolution (see Fig. 40.30). Pseudo-deconvolution methods can correct for sym-... 

Computer software codes are available to deconvolute PIXE spectra and to calculate peak areas with accuracy, so that absolute amounts of elements present in the specimen may be derived. With a beam of 5 mm diameter incident on a thin organic specimen on a thin backing foil, trace elements can be detected at picogram levels. The x-ray production cross-sections, absorption coefficients and the various... 

Experimentally, the EMD function p(q) can be reconstructed from a set of Compton profiles J qz ) s, and B( r) from the EMD. However, A Air) is not a direct experimental product. By combining the experimental B(r) with theoretical B aik (r), we need to derive a semiexperimental AB(r). Since the atomic image is very weak, many problems must be cleared in experimental resolution, in reconstruction (for example, selection of a set of directions and range of qzs), in various deconvolution procedures and so on. First of all, high resolution experiments are desirable. 

In an NMR analysis of the effects of /-irradiation induced degradation on a specific polyurethane (PU) elastomer system, Maxwell and co-workers [87] used a combination of both H and 13C NMR techniques, and correlated these with mechanical properties derived from dynamic mechanical analysis (DMA). 1H NMR was used to determine spin-echo decay curves for three samples, which consisted of a control and two samples exposed to different levels of /-irradiation in air. These results were deconvoluted into three T2 components that represented T2 values which could be attributed to an interfacial domain between hard and soft segments of the PU, the PU soft segment, and the sol... 

In microphase-separated systems, ESR spectra may consist of a superposition of two contributions, from nitroxides in both fast and slow-tumbling regimes. Such spectra provide evidence for the presence of two types of domains with different dynamics and transition temperatures. This case was detected for a HAS-derived nitroxide radical in heterophasic polyfacrylonitrile-butadiene-styrene) (ABS) as shown in Figure 5, the fast and slow components in the ESR spectrum measured represent nitroxide radicals located in butadiene-rich (B-rich) and styrene/acrylonitrile-rich (SAN-rich) domains, respectively [40]. These two components were determined by deconvoluting the ESR spectrum of HAS-NO measured at 300 K. 

To derive S(t), a deconvolution procedure is required. The transducer function is easily obtained by running an experiment where the photoactive species decays with x 1/v, which is the case of the photoacoustic calibrants mentioned. Recall... 

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